Question 1199626
<br>
Note for future reference: It is standard to use "^" (shift-6) to represent exponentiation.  The equation in your problem is (y^2)/4 - (x^2)/9 = 1.<br>
The equation has both x^2 and y^2 terms, with opposite signs, and it contains no xy term.  So the graph is a hyperbola that opens either right and left (option B -- "to the sides") or up and down (option F).<br>
We need to determine which of those options is the right one.<br>
In my experience, a lot of references teach a rule: if the x^2 term is positive, the hyperbola opens in the x direction (right and left); if the y^2 term is positive, the hyperbola opens in the y direction (up and down).<br>
That works... but if a student just memorizes a rule and then forgets the rule or gets confused about what the rule says, then he will have trouble finding the answer.<br>
I recommend using a method that uses logical reasoning rather than memorization.<br>
What I do when I see your equation is suppose that y=0; then the equation says {{{-x^2/9=1}}}, which makes x^2 negative, which is impossible.  That means y can't be zero -- and that means the hyperbola opens up and down.<br>
Using logical reasoning rather than a memorized rule helps the student UNDERSTAND why the graph opens in the direction it does; and that makes the student enjoy the math more.<br>
ANSWER: up and down -- option F<br>